The graph of f(x) = a/x + q is given below. A is the point (-4; 3) and B is the point (2; 6) while C is (x; y) which is the x-intercept, q is asymptotes.

1.1.1 Determine the value(s) of a and q. The general form of the function is f(x) = (a)/(x) + q. The graph passes through point A (-4, 3) and point B (2, 6). Substitute these points into the equation: Step 1: Substitute point A (-4, 3) into the equation. 3 = (a)/(-4) + q 12 = -a + 4q (Equation 1) Substitute point B (2, 6) into the equation. 6 = (a)/(2) + q 12 = a + 2q (Equation 2) Add Equation 1 and Equation 2 to eliminate a. (12) + (12) = (-a + 4q) + (a + 2q) 24 = 6q q = (24)/(6) q = 4 Substitute q=4 into Equation 2 to find a. 12 = a + 2(4) 12 = a + 8 a = 12 - 8 a = 4 The values are a=4 and q=4. a=4, q=4 1.1.2 Show that the equation of f is f(x) = (4)/(x) + 4. Step 2: Substitute the values of a and q found in 1.1.1 into the general equation f(x) = (a)/(x) + q. f(x) = (4)/(x) + 4 This matches the given equation. 1.1.3 The range of f. Step 3: The function is f(x) = (4)/(x) + 4. For a hyperbola of the form f(x) = (a)/(x) + q, the horizontal asymptote is y=q. The range includes all real numbers except the value of the horizontal asymptote. Since q=4, the horizontal asymptote is y=4. The range of f is all real numbers except y=4. y R, y ≠ 4 1.1.4 The domain of f. Step 4: For the function f(x) = (4)/(x) + 4, the denominator cannot be zero. Therefore, x ≠ 0. The domain of f is all real numbers except x=0. x R, x ≠ 0 1.1.5 The coordinates of C. Step 5: C is the x-intercept, which means f(x) = 0. Set the equation f(x) = (4)/(x) + 4 to 0. 0 = (4)/(x) + 4 -4 = (4)/(x) Multiply both sides by x. -4x = 4 x = (4)/(-4) x = -1 The coordinates of C are (-1, 0). C(-1, 0) That's 2 down. 3 left today — send the next one.